P-adic Functions - Part 1

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1 P-adic Functions - Part 1 Nicolae Ciocan Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant functions. We will look at some interesting properties of such functions. Definition 1. A function f : Z p Q p is called continuous at the point a Z p if ɛ > 0 there δ > 0 such that x a p < δ implies f(x) f(a) p < ɛ, x Z p. A function f : Z p Q p is continuous if it is continuous at all points a Z p. A function f : Z p Q p is uniformly continuous if ɛ > 0, δ > 0 such that x y p < δ implies f(x) f(y) p < ɛ, x, y Z p. Example 1. Since the space Z p is totally disconnected, the charasteristic function of any ball U Z p : { 0, if x U ξ U (x) : 1, if x Z p \U is continuous. This is clear since both the ball U and its complement Z p \U are open. Definition 2. A function f : Z p Q p is called locally constant if x Z p there exists an open neightbourhood U x of x (e.g. a ball of radius p m for some m N centered at x, U x = {y Z p x y p < p m }) such that f is constant on U x. Example 2. For α Z p we have its p-adic expension α = α 0 + αp α n p n +..., α n Z, 0 < α n (p 1) 1

2 Define f n : Z p Q p f n (α) = α n, n These functions are locally constant, because f n remains unchanged if we replace α by any β with β α p < 1 p n Furthermore we can extend these functions to the space Q p. Theorem 1. Locally constant functions are continuous. Proof. Result follows directly from Definition 2. Theorem 2. Let f : Z p Q p be a locally constant function. Then Z p can be written as the union k Z p = i=1 of finitely many disjoint balls such that the function f is constant on each of these balls. In particular, the set {f(x) x Z p } of all values assumed by f on Z p has only finitely many distinct values. U xi Proof. Let U x be defined like in Definition 2. These balls form a cover of Z p. By compactness of Z p, this cover contains a finite subcover U x1, U x2,..., U xk. Furthermore, we know that two balls in a ultra-metric space are either disjoint or contained in one another. So if we delete all the balls which lie inside other balls, we obtain a cover of Z p by disjoint balls. Theorem 3. Any locally constant function in Z p is uniformly continuous. Proof. Let k Z p = i=1 be a partition on Z p into disjoint balls as in Theorem 2. Let p m i be the radius of U xi, i = 1,2,...,k, m = max(m i ) and δ = p m. We prove that for ɛ > 0 the chosen δ works: Suppose that x y p < δ = p m. Since x U xi for some i and each point of the ball is its center, we may assume x = x i. Then x i y p < p m p m i which means that f(y) = f(x i ) = f(x). U xi 2

3 Definition 3. Let E Z p, not necessarily compact. A function f : E Q p is called a step function on E if there exists a positive integer t such that f(x) = f(x 0 ) for all x, x 0 E with x x 0 p p t. The smallest integer t for which this property holds is called the order of f. Remark: It is clear from the definition that a step function is uniformly continuous and also locally constant on E. On Z p (or on any compact set) it also holds, that any locally constant function is a step function. Remark: In p-adic analisys, the step functions have a really surprising property. But first, for a positive integer t, we want to construct an explicit partition of E Z p. Let N t = {0, 1,..., p t 1}. For each x Z p we write its canonical expansion : and we define It follows that Let x = x 0 + x 1 p x t 1 p t 1 + x t p t +... N x = x 0 + x 1 p x t 1 p t 1. N x N t and x N x p p t U(N, t) = {x Z p x N x p p t < p t+1 }. For each N N p we define E(N) = E U(N, t) We already seen that any x Z p belongs to some U(N, t) and since for any N, M N t we have N M p > p t, it follows that the balls U(N, t) are disjoint. Therefore: is a partition of E. E = p t 1 N=0 E(N) Theorem 4. Any step function on N or Z p is periodic. Proof. Let E = N or E = Z p and f : E Q p be a step function of order t. Consider the partition o E: E = p t 1 N=0 E(N) 3

4 If x, y E(N) we have x y p = (x N) + (N y) p p t by the strong triangle inequality and hence f(x) = f(y). Notice that if x E(N), then x+p t E(N). Therefore which means f is periodic. f(x + p t ) = f(x) for x E, Remark: In real analysis continuous functions on closed intervals can be uniformly approximated by step functions. In p-adic analysis we have a similar result and additionally, p-adic step functions are continuous. Theorem 5. Let E = N or E = Z p. A function f : E Q p is uniformly continuous on E if and only if for every positive integer s there exists another positive integer t=t(s) and a step function S : E Q p of order at most t such that:. f(x) S(x) p p s, x E Proof. = Assume that f is uniformly continuous on E. Choose the two positive integers s and t=t(s) such that: f(x) f(x 0 ) p p s for x, x 0 E with x x 0 p p t For x we have its p-adic expansion: and we define Define the function S : E Q p by x = x 0 + x 1 p x t 1 p t 1 + x t p t +... N x = x 0 + x 1 p x t 1 p t 1. S(x) = f(n x ) if x E. S is a step function of order at most t. Then we get : f(x) S(x) p = f(x) f(n x ) p p s, which is what we want. = Choose f and S like in our assumption. If x 0 satisfies: x x 0 p p t. 4

5 Then we get S(x) = S(x 0 ) f(x) S(x) p p s f(x 0 ) S(x 0 ) p p s resulting: f(x) f(x 0 ) p = (f(x) S(x)) (f(x 0 ) S(x 0 )) p p s from which follows that f is uniformly continuous. 2 Continuous and uniformly continuous functions In the following let E Z p and let x 0 E be an accumulation point of E. Theorem 6. Let f : E Q p, g : E Q p. (1) f is continuous at x 0 E if and only if for every sequence {x n } satisfying lim x x n = x 0 we have lim x f(x n) = f(x 0 ). (2) If f and g are continuous at x 0 E the so are f + g, f g, fg, and if g(x 0 ) 0 then so is f/g. Proof. The proof is very similar to the real case. Example 3. Let f : N Q p be defined as f(x) = 1 x c with c Z p. If c / N, then the denominator does not vanish and therefore, by the above theorem, f is continuous on N. However f is not bounded on N - since c is a p-adic integer, we can find elements in N for which x c p is arbitrarily small, hence f(x) p is arbitrarily large. If c N then f is not continuous at the point c. Theorem 7. Every continuous function f : Z p Q p is uniformly continuous and bounded on Z p. Proof. The theorem followings directly from compactness of Z p. 5

6 Remark: Similar to real analysis we have the following extension result. Theorem 8. Let E be a subset of Z p and let Ē be its closure. Let f : E Q p be a uniformly continuous on E. Then there exists a unique function F : Ē Q p uniformly continuous and bounded on Ē such that F (x) = f(x), for x E. Proof. Let y Ē. Then there exists a sequence {x n} in E such that x n y for n Suppose y / E (if y E it s clear). Since f is uniformly continuous on E, for any positive integer s there exists another positive integer t=t(s) such that (*) f(x) f(x 0 ) p p s for x, x 0 E with x x 0 p p t is satisfied. Because x n y there is an integer N = N(t) such that Therefore fo n, m N we also have and hence x n y p p t for n N. x m x n p = (x m y) (x n y) p p t, f(x m ) f(x n ) p p s. This means that {f(x n )} is a p-adic Cauchy sequence, and since Q p is complete, it s limit L = lim n f(x n ) Q p. It is easy to see that the limit does not depend on the chosen sequence x n y. Indeed, let {z n } be another sequnce such that z n y. Then {x n z n } will be a null sequence and by the uniform continuity of f, {f(x n ) f(z n )} is also a null sequence, but this implies that Thus the function F : Ē Q p given by L = lim n f(z n). F (y) = lim n f(x n) whenever y Ē and y = lim n x n E and x n E is well defined. Now we show that the function F is uniformly continuous on Ē. Let y and y 0 be two points in Ē satisfying y y 0 p p t. Choose x and x 0 in E such that: x y p p t 6

7 It follows that x 0 y 0 p p t f(x) f(y) p p s f(x 0 ) f(y 0 ) p p s x x 0 p = (x y) + (y y 0 ) (x 0 y 0 ) p p t hence by (*) we have f(x) f(x 0 ) p p t. Therefore: F (y) F (y 0 ) p = (f(x) F (y)) + (f(x) f(x 0 )) + (f(x 0 ) F (y 0 )) p p s proving the uniform continuity of F on Ē. Finally, F is bounded on Ē. Indeed, otherwise there would exist an infinite sequence {y n } Ē such that lim F (y n ) p =. Since E and hence Ē are subsets of a compact set Z p, there exists a subsequence {y rn } such that the limit y 0 = lim n y r n exists. Since all points are in Ē and since Ē is a closed set, we have y 0 Ē. Now F is uniformly continuous on Ē and therefore continuous at y 0. But this implies lim n F (y r n ) = F (y 0 ) contrary to lim n F (y n ) p =. To prove the uniqueness of F, we assume that there is a second function G with the same proprieties. Then F-G is uniformly continuous on Ē and identically 0 on E. Since E is dense in Ē, by continuity F G is also identically 0 on Ē. Theorem 9. Let E Z p. If {f n } is a sequence of continuous functions on E. If f n f converges uniformly on E, then f is continuous on E. Proof. Similar to the real case. 7

8 P-ADIC ANALYSIS COMPARED WITH REAL CONTINUITY AND DIFFERENTIABILITY STEPHAN TORNIER 1. Points of Continuity When studying maps from a topological space X to itself one may ask whether there is a map that is continuous at every point of a given subset E X and discontinuous at every point of its complement E c. For instance, if E = X the identity will do. Choosing E = amounts to finding a map that is discontinuous at every point of X. In case X {R, Q p }, further distinguished subsets are E = Q, which in both cases is countable dense, and E = Q c. It turns out that for E {, Q c } such maps may be constructed whereas for E = Q it is impossible. Proposition 1.1. Let X {R, Q p } and E {, Q c }. There is a map from X to itself whose set of points of continuity is E. Proof. Let E =. The map χ Q will do for both R and Q p. For Q p note that both Q and Q c are dense in Q p whence the preimage of an open set containing either 0 Q p or 1 Q p but not both cannot contain an open set. Next, let E = Q c. For X = R consider the Thomae map t : R R defined by 1 if x = 0 1 t : x q if x = p q Q is reduced. 0 x Q c Its set of points of continuity is exactly Q c. For X = Q p, consider f : x { 1 ν(x) 2 +1 x Q 0 x Q c where ν(x) denotes the index in the p-adic expansion of x at which the latter s period has repeated once. Then f is discontinuous at every point in x Q: we may choose a sequence (x n ) n N of non-rational p-adic numbers converging to x due to the fact that Q c is dense in Q p. Then 0 = lim n f(x n ) f(x) > 0. However, f is continuous at every point of x Q c : if (x n ) n is a sequence of p-adic numbers converging to x then the image sequence (f(x nm )) m of its subsequence (x nm ) m of rational numbers converges to zero as the index up to which the p-adic expansions of x and x nm coincide grows as m tends to infinity. Hence ν xnm as well. The impossibility of such a construction for E = Q may be viewed as a consequence of the Baire Category Theorem and is elaborated on subsequently. Recall [2] that complete metric spaces are Baire spaces. In particular, a complete metric space cannot be expressed as a countable union of nowhere dense sets. Proposition 1.2 (E = Q). Let X {R, Q p }. There is no map from X to itself whose set of points of continuity is Q. Date: November 24,

9 2 STEPHAN TORNIER The proof of proposition 1.2 is based on the following property of the set of points of continuity. Definition 1.3. A subset of a topological space is of type G δ if it is a countable intersection of open sets, and of type F σ if it is countable union of closed sets. As a mnenomic device use e.g. G Gebiet, δ Durchschnitt, F fermé and σ sum. Proposition 1.4. Let X and Y be metric spaces. The set of points of continuity of a map f : X Y is of type G δ. Its set of points of discontinuity is of type F σ. Proof. It suffices to prove one assertion. For any subset A of X, define the oscillation of f on A by ω f (A) := sup{d(f(x),f(y)) x,y A} R { } and the oscillation of f at a point x X by ω f (x) := lim ε 0 ω f (B(x,ε)) if the limit exists. Then f is continuous at x X if and only if ω f (x) = 0 whence its set of points of continuity equals {x X ω f (x) = 0} = n N{x X ω f (x) < 1/n}. The assertion follows as the sets {x X ω f (x) < 1/n}, n N are open. Baire s Theorem for complete metric spaces and proposition 1.4 now imply the following result of which proposition 1.2 appears as an immediate corollary. Proposition 1.5. Let X be a complete metric space and Y a metric space. Then there is no map f : X Y whose set of points of continuity E is countable dense. Proof. Assume such a map f exists. Then its set of points of discontinuity is of type F σ, i.e. X E = n NF n for closed sets F n,n N. Then each F n is nowhere dense. Otherwise, X E would contain an open ball contradicting the assumption that E is dense in X. But E is the union of its points which are nowhere dense subsets of X as well. Hence X is a countable union of nowhere dense sets in contradiction to being a Baire space. 2. Differentiability In this section we elaborate on whether Rolle s Theorem, the Mean Value Theorem and the Inverse Function Theorem from elementary calculus carry over the p-adic setting, see Katok [1, Ch. 4.4]. Theorem 2.1 (Rolle s Theorem). Let f : [a,b] R be continuous. If f is differentiable on (a,b) and f(a) = f(b) there is some ξ (a,b) such that f (ξ) = 0. Apart from the fact that Q p cannot be made into an ordered fields whence there is no notion of interval there is no hope for a version of Rolle s theorem to hold due to the following example. Proposition 2.2. There is a differentiable map f : Z p Q p satisfying both f(0) = f(1) and f (x) 0 x Z p. Proof. Let f : Z p Q p be defined by x x p x. Then f(0) = 0 = f(1) but f (x) = px p 1 1 whence f (x) p Z p 1 x Z p. Hence f (x) 0 x Z p.

10 P-ADIC ANALYSIS COMPARED WITH REAL CONTINUITY AND DIFFERENTIABILITY 3 In real analysis, an important consequence of Rolle s Theorem is the Mean Value Theorem which implies a criterion for a map to be constant. Corollary 2.3 (Mean Value Theorem). Let f : [a,b] R be continuous. If f is differentiable on (a,b) there is some ξ (a,b) such that f(b) f(a) b a = f (ξ). Corollary 2.4. Let f : (a,b) R be differentiable with f 0. Then f is constant. As above, corollaries 2.3 and 2.4 would require an ordering but fail in a much stronger sense in the p-adic setting. Proposition 2.5. There is an injective, differentiable map f : Z p Z p Q p whose derivative vanishes identically. Proof. Let f : Z p Z p Q p be defined by For x,y Z p we obtain f : a k p k a k p 2k k=0 k=0 f(x) f(y) p = p ord(f(x) f(y)) = p 2 ord(x y) = x y 2 p which implies injectivity as well as vanishing derivative. The Inverse Function Theorem fails in a rather devastating sense as well. Theorem 2.6 (Inverse Function Theorem). Let U R be open and let f : U R be C 1. If f (a) 0 for some a U then f is a local diffeomorphism at a. Proposition 2.7. There is a continuously differentiable map f : Z p Q p with nowhere vanishing derivative which is injective in no neighbourhood of 0 Z p. Proof. For each n N, let B n := B p 2n(p n ) = {x Z p x p n p < p 2n } Z p. Since x B n implies x p = p n we have B n B m = whenever n m. Define { x p 2n if x B n f : x x if x Z p n N B. n Then f(p n ) = p n p 2n as p n B n. But p n p 2n is contained in no B m. Therefore f(p n p 2n ) = p n p 2n and f is injective in no neighbourhood of 0 Z p. As to differentiability, consider g(x) := x f(x). We aim to show that g exists and vansishes everywhere in which case f (x) 1. For a start, g(x) is locally constant on Z p {0} and hence has the derivative g (x) = 0 on Z p {0}. It remains to check that g (0) = 0. But whence the assertion. f(x) f(0) x = p { p 2n p x p = p n x B n 0 x Z p n N B n However, there is a stronger notion of differentiability than C 1 that helps restoring injectivity.

11 4 STEPHAN TORNIER Definition 2.8 (Strict differentiability). Let E Q p be non-empty without isolated points and let Φf : (E E) Q p, (x,y) f(x) f(y) x y where := {(x, x) x E}. Then f is called strictly differentiable at a E if (1) lim (x,y) (a,a) (Φf)(x,y) = f (a). Remark 2.9. It follows from definition 2.8 that any strictly differentiable map is continuously differentiable. The converse does not hold in the p-adic setting as the proof of proposition 2.7 shows: f(p n ) f(p n p 2n ) lim n p 2n = 0 1 = f (0). For continuously differentibale maps from R to itself however, the mean value theorem implies the existence of the limit (1). Proposition Let E Q p be non-empty without isolated points and let f : E Q p be strictly differentiable at a E. If f (a) 0 then f is injective on some neighbourhood of a. Proof. We will construct a neighbourhood of a Z p on which f/f (a) is an isometry, thus in particular injective. By definition of strict differentiability there exists some δ > 0 such that for all x y Z p which satisfy x a p δ and y a p δ we have f(x) f(y) f (a) x y f (a) p. p The isosceles triangle inequality then implies f(x) f(y) x y = f (a) p p whence B δ (a) serves as the desired neighbourhood. References 1. S. Katok, p-adic Analysis Compared with Real, American Mathematical Society, B. v. Querenburg, Mengentheoretische Topologie, Springer, 1979.